Determine the set of all points $(x,y)$ in two-dimensional cartesian coordinate system such that \begin{align*}0\le &\,x\le\frac{\pi}{2}, \\ \sqrt{1-\sin 2x}-\sqrt{1+\sin 2x}\le &\,y\le\sqrt{1-\cos2x}-\sqrt{1+\cos2x}.\end{align*} Draw a picture of the set.

Find all possible real values of $a$ for which the system of equations \[\{\begin{array}{cc}x +y +z=0\\\text{ } \\ xy+yz+azx=0\end{array}\] has exactly one solution.

Points $K$ and $L$ are taken on the sides $BC$ and $CD$ of a square $ABCD$ so that $\angle AKB = \angle AKL$. Find $\angle KAL$.

The figure shows a game board with $16$ squares. At the start of the game, two cars are placed in different squares. Two players $A$ and $B$ alternately take turns, and A starts. In each turn, the player chooses one of the cars and moves it one or more squares to the right. The left-most car may never overtake or land on the same square as the right-most car. The first player which is unable to move loses. [img]https://cdn.artofproblemsolving.com/attachments/1/b/8d6f40fac4983d6aa9bd076392c91a6d200f6a.png[/img] (a) Prove that A can win regardless of how $B$ plays, if the two cars start as shown in the figure. (b) Determine all starting positions in which $B$ can win regardless of how $A$ plays.

Find the max. value of $ M$,such that for all $ a,b,c>0$: $ a^{3}+b^{3}+c^{3}-3abc\geq M(|a-b|^{3}+|a-c|^{3}+|c-b|^{3})$

For certain $a,b,c$ holds: $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ Prove that for all odd $n$ holds, $$\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}.$$

A square sheet of paper that measures $18$ cm on a side has corners labeled $A$, $B$, $C$, and $D$ in clockwise order. Point $B$ is folded over to a point $E$ on $\overline{AD}$ with $DE=6$ cm and the paper is creased. When the paper is unfolded, the crease intersects side $\overline{AB}$ at $F$. Find the number of centimeters in $FB$.

Let $ \{D_1, D_2, ..., D_n \}$ be a set of disks in the Euclidean plane. Let $ a_ {i, j} = S (D_i \cap D_j) $ be the area of $ D_i \cap D_j $. Prove that $$ \sum_ {i = 1} ^ n \sum_ {j = 1} ^ n a_ {i, j} x_ix_j \geq 0 $$ for any real numbers $ x_1, x_2, ..., x_n $.

Given the vector spaces $V,W$ with coefficients over a field $K$ and function $ \phi :V\to W$ satisfying the relation : $$\varphi(\lambda x+y)= \lambda \varphi(x)+\phi (y)$$ for all $x,y \in V, \lambda \in K$. Such a function is called linear. Let $L\varphi=\{x\in V/\varphi(x)=0\}$ , and$M=\varphi(V)$ , prove that : (i) $L\varphi$ is subspace of $V$ and $M$ is subspace of $W$ (ii) $L\varphi={O}$ iff $\varphi$ is $1-1$ (iii) Dimension of $V$ equals to dimension of $L\varphi$ plus dimension of $M$ (iv) If $\theta : \mathbb{R}^3\to\mathbb{R}^3$ with $\theta(x,y,z)=(2x-z,x-y,x-3y+z)$, prove that $\theta$ is linear function . Find $L\theta=\{x\in {R}^3/\theta(x)=0\}$ and dimension of $M=\theta({R}^3)$.

Let $a$, $b$, $c$, $d$, $e$, $f$ and $g$ be seven distinct positive integers not bigger than $7$. Find all primes which can be expressed as $abcd+efg$

In a convex quadrilateral $ABCD$, both $\angle ADB=2\angle ACB$ and $\angle BDC=2\angle BAC$. Prove that $AD=CD$.

There are $n \ge 6$ green points in the plane, such that no $3$ of them are collinear. Suppose further that $6$ of these points are the vertices of a convex hexagon. Prove that there are $5$ green points that form a pentagon that does not contain any other green point inside.

Alicia and Bob take turns writing words on a blackboard. The rules are as follows: a) Any word that has been written cannot be rewritten. b) A player can only write a permutation of the previous word, or can simply simply remove one letter (whatever you want) from the previous word. c) The first person who cannot write another word loses. If Alice starts by typing the word ''Olympics" and Bob's next turn, who, do you think, has a winning strategy and what is it?

Let $u_k$, $v_k$, $a_k$ and $b_k$ be non-negative real sequences such as $u_k > a_k$ and $v_k > b_k$, where $k = 1, 2,\cdots , n$. If $0 < m_1 \leq u_k \leq M_1$ and $0 < m_2 \leq v_k \leq M_2$, then $$\sum \limits_{k=1}^n(lu_kv_k-a_kb_k)\geq \left(\sum \limits_{k=1}^n\left(u_k^2-a_k^2\right)\right)^\frac{1}{2}\left(\sum \limits_{k=1}^n\left(v_k^2-b_k^2\right)\right)^\frac{1}{2}$$where$$l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}$$

If $a,b\neq 0$ are real numbers such that $a^2b^2(a^2b^2+4)=2(a^6+b^6)$ , then show that $a,b$ can’t be both of them rational .

If the sum of all positive divisors (including itself) of a positive integer $n$ is $2n$, then $n$ is called a perfect number. For example, the sum of the positive divisors of 6 is $1 + 2 + 3 + 6 = 2 \times 6$, hence 6 is a perfect number. Prove: There does not exist a perfect number of the form $p^a q^b r^c$, where $a, b, c$ are positive integers, and $p, q, r$ are odd primes.

Let $X_1$, $X_2$, ..., $X_{2012}$ be chosen independently and uniformly at random from the interval $(0,1]$. In other words, for each $X_n$, the probability that it is in the interval $(a,b]$ is $b-a$. Compute the probability that $\lceil\log_2 X_1\rceil+\lceil\log_4 X_2\rceil+\cdots+\lceil\log_{1024} X_{2012}\rceil$ is even. (Note: For any real number $a$, $\lceil a \rceil$ is defined as the smallest integer not less than $a$.)

[i](6 pts)[/i] Find all positive integers $n \ge 3$ such that there exists a permutation $a_{1}, a_{2}, \dots, a_{n}$ of $1, 2, \dots, n$ such that $a_{1}, 2a_{2}, \dots, na_{n}$ can be rearranged into an arithmetic progression.

a sequence $(a_n)$ $n$ $\geq 1$ is defined by the following equations; $a_1=1$, $a_2=2$ ,$a_3=1$, $a_{2n-1}$$a_{2n}$=$a_2$$a_{2n-3}$+$(a_2a_{2n-3}+a_4a_{2n-5}.....+a_{2n-2}a_1)$ for $n$ $\geq 2$ $na_{2n}$$a_{2n+1}$=$a_2$$a_{2n-2}$+$(a_2a_{2n-2}+a_4a_{2n-4}.....+a_{2n-2}a_2)$ for $n$ $\geq 2$ find $a_{2020}$

Let $f:(-1,1)\to \mathbb{R}$ be a twice differentiable function such that $$2f’(x)+xf''(x)\geqslant 1 \quad \text{ for } x\in (-1,1).$$ Prove that $$\int_{-1}^{1}xf(x)dx\geqslant \frac{1}{3}.$$ [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karim Rakhimov, Scuola Normale Superiore and National University of Uzbekistan[/i]

Let $ ABC$ be an equilateral triangle. Let $ \Omega$ be its incircle (circle inscribed in the triangle) and let $ \omega$ be a circle tangent externally to $ \Omega$ as well as to sides $ AB$ and $ AC$. Determine the ratio of the radius of $ \Omega$ to the radius of $ \omega$.

For all positive real numbers $a, b, c$ satisfying $ab+bc+ca \leq 1,$ prove that \[ a+b+c+\sqrt{3} \geq 8abc \left(\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right) \]

[b]2.[/b] Let $n \geq 3$ be an integer. Prove that for all integers $k$, with $1 \leq k \leq \binom{n}{2}$, there exists a set $A$ with $n$ distinct positive integer elements such that the set $B = \{\gcd(x, y): x, y \in A, x \neq y \}$ (gotten from the greatest common divisor of all pairs of distinct elements from $A$) contains exactly $k$ distinct elements.

Dua has all the odd natural numbers less than 20. Asija has all the even numbers less than 21. They play the following game. In each round, they take a number from each other and after every round, they may fix two or more consecutive numbers so that their opponent cannot take these fixed numbers in the next round. The game is won by the player who attains 10 consecutive numbers first. Does either player have a winning strategy?

Let be a $ 5\times 5 $ complex matrix $ A $ whose trace is $ 0, $ and such that $ I_5-A $ is invertible. Prove that $ A^5\neq I_5. $